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Properties

Label 2-208-13.12-c9-0-16
Degree $2$
Conductor $208$
Sign $0.148 - 0.988i$
Analytic cond. $107.127$
Root an. cond. $10.3502$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 182.·3-s − 909. i·5-s + 3.89e3i·7-s + 1.36e4·9-s − 4.20e4i·11-s + (−1.53e4 + 1.01e5i)13-s − 1.66e5i·15-s − 4.05e5·17-s + 7.33e5i·19-s + 7.11e5i·21-s + 2.38e6·23-s + 1.12e6·25-s − 1.09e6·27-s − 6.21e6·29-s − 1.75e6i·31-s + ⋯
L(s)  = 1  + 1.30·3-s − 0.650i·5-s + 0.613i·7-s + 0.694·9-s − 0.865i·11-s + (−0.148 + 0.988i)13-s − 0.846i·15-s − 1.17·17-s + 1.29i·19-s + 0.798i·21-s + 1.77·23-s + 0.576·25-s − 0.397·27-s − 1.63·29-s − 0.342i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(107.127\)
Root analytic conductor: \(10.3502\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :9/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.632087681\)
\(L(\frac12)\) \(\approx\) \(2.632087681\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (1.53e4 - 1.01e5i)T \)
good3 \( 1 - 182.T + 1.96e4T^{2} \)
5 \( 1 + 909. iT - 1.95e6T^{2} \)
7 \( 1 - 3.89e3iT - 4.03e7T^{2} \)
11 \( 1 + 4.20e4iT - 2.35e9T^{2} \)
17 \( 1 + 4.05e5T + 1.18e11T^{2} \)
19 \( 1 - 7.33e5iT - 3.22e11T^{2} \)
23 \( 1 - 2.38e6T + 1.80e12T^{2} \)
29 \( 1 + 6.21e6T + 1.45e13T^{2} \)
31 \( 1 + 1.75e6iT - 2.64e13T^{2} \)
37 \( 1 - 1.58e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.90e7iT - 3.27e14T^{2} \)
43 \( 1 + 1.39e7T + 5.02e14T^{2} \)
47 \( 1 - 1.76e7iT - 1.11e15T^{2} \)
53 \( 1 - 1.03e8T + 3.29e15T^{2} \)
59 \( 1 - 1.34e8iT - 8.66e15T^{2} \)
61 \( 1 + 4.27e7T + 1.16e16T^{2} \)
67 \( 1 - 1.49e8iT - 2.72e16T^{2} \)
71 \( 1 + 2.27e8iT - 4.58e16T^{2} \)
73 \( 1 + 1.48e8iT - 5.88e16T^{2} \)
79 \( 1 + 8.27e7T + 1.19e17T^{2} \)
83 \( 1 - 3.33e8iT - 1.86e17T^{2} \)
89 \( 1 + 2.12e8iT - 3.50e17T^{2} \)
97 \( 1 - 8.43e8iT - 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.01965961096865464580567297484, −9.524613576985229861925398878357, −8.848791087185913683477228905532, −8.397316746829712399372863892590, −7.13009344445947106462914624548, −5.81664731875899355107972292078, −4.54109384294793688789696951587, −3.38660574788063313423989501534, −2.35997806757413815011567594693, −1.28133040448793233315752057329, 0.44006508480098164145295608493, 2.09338467588672988386537176634, 2.89389786747188611434100411652, 3.89839697494776583879644271174, 5.16765190236204910738858783137, 7.06405920839117394379559238197, 7.27205031299881035794077547901, 8.685962329237829568108706815498, 9.331980818447253039044621451946, 10.52797538270876405944434919595

Graph of the $Z$-function along the critical line